Subgroups — #36442473

Order 22784.

426 conjugacy classes of subgroups.

The subgroup lattice has 314 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#15668004	C₂	1	1	yes	maximal, characteristic
2	#26020395	C₂	1	1	yes	maximal, characteristic
3	#26018076	C₂	1	2	yes	maximal
4	#26018074	C₂	1	2	yes	maximal
5	C₄×C₄⋊C₈×C₈₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#15668004	C₄	1	1	yes	characteristic
7	#19151496	C₂²	1	1	yes	characteristic
8	#19151496	C₂²	1	1	yes	characteristic
9	#19151482	—	2	1	no	—
10	#19151482	—	2	1	no	—
11	#19151464	C₄	1	1	yes	characteristic
12	C₄×#13083553	—	2	1	no	—
13	C₄×#13083553	—	2	1	no	—
14	C₄×C₄⋊Dic₈₉	C₄	1	1	yes	characteristic
15	C₄×C₄⋊Dic₈₉	C₄	1	1	yes	characteristic
16	#19151353	—	2	1	no	—
17	#19151353	—	2	1	no	—
18	#19151353	C₂²	1	2	yes	—
19	C₂×C₄⋊C₈×C₈₉	—	2	1	no	—
20	C₂×C₄×C₈×C₈₉	—	2	1	no	—
21	C₄³×C₈₉	C₂²	1	1	yes	characteristic
22	#336×C₈₉	C₂²	1	2	yes	—
23	C₂²×#10623494	—	2	2	no	—
24	#15668004	—	2	2	no	—
25	#15668004	C₈	1	4	yes	complemented
26	C₂×C₄⋊Dic₈₉	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄⋊Dic₈₉	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄⋊Dic₈₉	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄⋊Dic₈₉	C₂×C₄	1	1	yes	characteristic
30	C₂×#13083553	—	2	1	no	—
31	C₂×#13083553	—	2	1	no	—
32	C₂×#13083553	—	2	1	no	—
33	C₂×#13083553	—	2	1	no	—
34	C₂×C₄×Dic₈₉	—	2	1	no	—
35	C₂×C₄×Dic₈₉	—	2	1	no	—
36	#15667919	—	2	1	no	—
37	#15667919	—	2	1	no	—
38	#15667919	—	2	1	no	—
39	#15667919	—	2	1	no	—
40	#15667919	—	2	1	no	—
41	#15667919	—	2	1	no	—
42	C₂²×C₈×C₈₉	—	2	2	no	—
43	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
46	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
47	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
48	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
49	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
50	C₄⋊C₈×C₈₉	—	2	4	no	—
51	C₄×C₈×C₈₉	—	4	2	no	—
52	C₂²×Dic₈₉	—	2	1	no	—
53	C₂²×Dic₈₉	—	2	1	no	—
54	C₂²×Dic₈₉	—	2	1	no	—
55	C₂²×Dic₈₉	—	2	1	no	—
56	C₂×#10623494	—	4	2	no	—
57	C₂×#10623494	—	4	2	no	—
58	C₂×#10623494	—	4	2	no	—
59	C₄⋊Dic₈₉	—	2	1	no	—
60	C₄⋊Dic₈₉	—	2	1	no	—
61	C₄⋊Dic₈₉	—	4	1	no	—
62	C₄⋊Dic₈₉	—	4	1	no	—
63	C₄⋊Dic₈₉	C₂×C₈	1	2	yes	—
64	C₄⋊Dic₈₉	C₂×C₈	1	2	yes	—
65	#13083553	—	2	4	no	—
66	#13083553	—	2	4	no	—
67	#13083553	—	4	2	no	—
68	#13083553	—	4	1	no	—
69	#13083553	—	4	1	no	—
70	C₄×Dic₈₉	—	4	2	no	—
71	C₄×Dic₈₉	—	4	2	no	—
72	C₂²×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
73	C₂²×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
74	C₂²×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
75	C₂²×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
76	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
77	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
79	C₂×C₈×C₈₉	—	2	4	no	—
80	C₂×C₈×C₈₉	—	4	2	no	—
81	C₂×C₈×C₈₉	—	4	2	no	—
82	C₄²×C₈₉	—	2	2	no	—
83	C₄²×C₈₉	—	2	2	no	—
84	C₄²×C₈₉	—	2	2	no	—
85	C₄²×C₈₉	—	2	1	no	—
86	C₄²×C₈₉	—	2	1	no	—
87	C₄²×C₈₉	—	4	1	no	—
88	C₄²×C₈₉	—	4	1	no	—
89	C₄²×C₈₉	C₂×C₈	1	2	yes	—
90	C₄²×C₈₉	M₁₆	1	2	yes	—
91	C₂³×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
92	C₂×C₄×C₈₉	—	2	1	no	—
93	C₂×C₄×C₈₉	—	2	1	no	—
94	C₂×C₄×C₈₉	—	2	1	no	—
95	C₂×C₄×C₈₉	—	2	1	no	—
96	C₂×C₄×C₈₉	—	2	1	no	—
97	C₂×C₄×C₈₉	—	2	1	no	—
98	C₂×C₄×C₈₉	—	2	1	no	—
99	C₂×C₄×C₈₉	—	2	1	no	—
100	C₂×C₄×C₈₉	—	2	1	no	—
101	C₂×C₄×C₈₉	—	2	1	no	—
102	C₂×C₄×C₈₉	—	2	1	no	—
103	C₂×C₄×C₈₉	—	2	1	no	—
104	C₂×C₄×C₈₉	—	2	1	no	—
105	C₂×C₄×C₈₉	—	2	1	no	—
106	C₂×C₄×C₈₉	C₂²⋊C₈	1	2	yes	—
107	C₂×C₄×C₈₉	C₂²⋊C₈	1	2	yes	—
108	C₂×C₄×C₈₉	#101	1	2	yes	—
109	C₂×C₄×C₈₉	#104	1	2	yes	—
110	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
111	C₂×C₄×C₈₉	C₂×M₁₆	1	1	yes	characteristic
112	C₂×C₄×C₈₉	#135	1	2	yes	—
113	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
114	C₂×C₄×C₈₉	#137	1	1	yes	characteristic
115	C₂×Dic₈₉	—	2	2	no	—
116	C₂×Dic₈₉	—	2	2	no	—
117	C₂×Dic₈₉	—	4	1	no	—
118	C₂×Dic₈₉	—	4	1	no	—
119	C₂×Dic₈₉	—	4	1	no	—
120	C₂×Dic₈₉	—	4	1	no	—
121	C₂×Dic₈₉	—	4	1	no	—
122	C₂×Dic₈₉	—	4	1	no	—
123	C₂×Dic₈₉	—	4	1	no	—
124	C₂×Dic₈₉	—	4	1	no	—
125	C₂×Dic₈₉	—	4	1	no	—
126	C₂×Dic₈₉	—	4	1	no	—
127	C₈×C₈₉	—	4	4	no	—
128	#10623494	—	8	2	no	—
129	C₂²×C₈₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
130	C₂²×C₈₉	#410	1	1	yes	characteristic
131	C₂²×C₈₉	C₂×#101	1	1	yes	characteristic
132	C₂²×C₈₉	#416	1	1	yes	characteristic
133	C₂²×C₈₉	#417	1	1	yes	characteristic
134	C₂²×C₈₉	C₂×#104	1	1	yes	characteristic
135	C₂²×C₈₉	#421	1	1	yes	characteristic
136	C₄×C₈₉	—	2	2	no	—
137	C₄×C₈₉	—	2	1	no	—
138	C₄×C₈₉	—	2	1	no	—
139	C₄×C₈₉	—	2	2	no	—
140	C₄×C₈₉	—	2	2	no	—
141	C₄×C₈₉	—	2	1	no	—
142	C₄×C₈₉	—	2	1	no	—
143	C₄×C₈₉	—	2	2	no	—
144	C₄×C₈₉	—	2	2	no	—
145	Dic₈₉	—	4	2	no	—
146	Dic₈₉	—	4	2	no	—
147	Dic₈₉	—	8	1	no	—
148	Dic₈₉	—	8	1	no	—
149	#7585	—	89	1	no	maximal, Sylow
150	C₂×C₈₉	#1461	1	1	yes	characteristic
151	C₂×C₈₉	#1476	1	1	yes	characteristic
152	C₂×C₈₉	#1477	1	1	yes	characteristic
153	C₂×C₈₉	#1509	1	1	yes	characteristic
154	C₂×C₈₉	#1521	1	1	yes	characteristic
155	C₂×C₈₉	#1528	1	1	yes	characteristic
156	C₂×C₈₉	#1552	1	1	yes	characteristic
157	C₄×#125	—	89	1	no	—
158	#1839	—	89	1	no	—
159	C₄×C₄⋊C₈	D₈₉	1	1	yes	characteristic
160	#1271	—	89	2	no	—
161	#1271	—	89	2	no	—
162	C₈₉	#7585	1	1	yes	characteristic, complemented, Sylow
163	C₂×#125	—	89	1	no	—
164	C₂×C₄⋊C₈	—	2	1	no	—
165	C₂×C₄×C₈	—	2	1	no	—
166	#389	—	89	1	no	—
167	#389	—	89	1	no	—
168	#387	—	178	1	no	—
169	#387	—	178	1	no	—
170	#382	—	89	1	no	—
171	C₄×C₄⋊C₄	—	89	1	no	—
172	C₄×C₄⋊C₄	—	89	1	no	—
173	C₄×C₄⋊C₄	—	178	1	no	—
174	C₄×C₄⋊C₄	—	178	1	no	—
175	C₄³	C₂×D₈₉	1	1	yes	characteristic
176	#336	—	89	2	no	—
177	#336	—	178	1	no	—
178	#336	—	178	1	no	—
179	#336	C₂×D₈₉	1	2	yes	—
180	C₂²×C₈	—	2	2	no	—
181	C₂²×C₈	—	178	2	no	—
182	#125	—	89	4	no	—
183	#125	—	178	2	no	—
184	C₂×C₄⋊C₄	—	89	1	no	—
185	C₂×C₄⋊C₄	—	89	1	no	—
186	C₂×C₄⋊C₄	—	89	1	no	—
187	C₂×C₄⋊C₄	—	89	1	no	—
188	C₂×C₄⋊C₄	—	178	1	no	—
189	C₂×C₄⋊C₄	—	178	1	no	—
190	C₂×C₄⋊C₄	—	178	1	no	—
191	C₂×C₄⋊C₄	—	178	1	no	—
192	C₂×C₄²	—	178	1	no	—
193	C₂×C₄²	—	178	1	no	—
194	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
195	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
196	C₂×C₄²	D₃₅₆	1	1	yes	characteristic
197	C₂×C₄²	D₃₅₆	1	1	yes	characteristic
198	C₂×C₄²	C₂²⋊D₈₉	1	1	yes	characteristic
199	C₂×C₄²	C₂²⋊D₈₉	1	1	yes	characteristic
200	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
201	C₄⋊C₈	—	2	4	no	—
202	C₄×C₈	—	4	2	no	—
203	#95	—	178	1	no	—
204	#95	—	178	1	no	—
205	#95	—	178	1	no	—
206	#95	—	178	1	no	—
207	#95	—	178	1	no	—
208	#95	—	178	1	no	—
209	C₂²×C₄	—	178	1	no	—
210	C₂²×C₄	—	178	1	no	—
211	C₂²×C₄	—	178	1	no	—
212	C₂²×C₄	—	178	1	no	—
213	C₂²×C₄	#13083551	1	1	yes	characteristic
214	C₂²×C₄	#13083551	1	1	yes	characteristic
215	C₂²×C₄	#13083551	1	1	yes	characteristic
216	C₂²×C₄	#13083551	1	1	yes	characteristic
217	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
218	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
219	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
220	C₂×C₈	—	2	4	no	—
221	C₂×C₈	—	4	2	no	—
222	C₂×C₈	—	4	2	no	—
223	C₂×C₈	—	356	2	no	—
224	C₂×C₈	—	356	2	no	—
225	C₂×C₈	—	356	2	no	—
226	C₄⋊C₄	—	89	2	no	—
227	C₄⋊C₄	—	89	2	no	—
228	C₄⋊C₄	—	356	1	no	—
229	C₄⋊C₄	—	356	1	no	—
230	C₄⋊C₄	—	356	2	no	—
231	C₄⋊C₄	—	356	1	no	—
232	C₄⋊C₄	—	356	1	no	—
233	C₄⋊C₄	—	178	1	no	—
234	C₄⋊C₄	—	178	1	no	—
235	C₄⋊C₄	—	178	4	no	—
236	C₄⋊C₄	—	178	4	no	—
237	C₄²	—	2	2	no	—
238	C₄²	—	2	1	no	—
239	C₄²	—	2	1	no	—
240	C₄²	—	2	2	no	—
241	C₄²	—	2	2	no	—
242	C₄²	—	4	1	no	—
243	C₄²	—	4	1	no	—
244	C₄²	—	356	2	no	—
245	C₄²	—	356	2	no	—
246	C₄²	C₈×D₈₉	1	2	yes	—
247	C₄²	#13083557	1	2	yes	—
248	C₂³	C₂×#13083551	1	1	yes	characteristic
249	C₂×C₄	—	2	1	no	—
250	C₂×C₄	—	2	1	no	—
251	C₂×C₄	—	2	1	no	—
252	C₂×C₄	—	2	1	no	—
253	C₂×C₄	—	2	1	no	—
254	C₂×C₄	—	2	1	no	—
255	C₂×C₄	—	2	1	no	—
256	C₂×C₄	—	2	1	no	—
257	C₂×C₄	—	2	1	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	2	1	no	—
260	C₂×C₄	—	2	1	no	—
261	C₂×C₄	—	2	1	no	—
262	C₂×C₄	—	2	1	no	—
263	C₂×C₄	—	356	1	no	—
264	C₂×C₄	—	356	1	no	—
265	C₂×C₄	—	356	1	no	—
266	C₂×C₄	—	356	1	no	—
267	C₂×C₄	—	356	1	no	—
268	C₂×C₄	—	356	1	no	—
269	C₂×C₄	—	356	1	no	—
270	C₂×C₄	—	356	1	no	—
271	C₂×C₄	—	356	1	no	—
272	C₂×C₄	—	356	1	no	—
273	C₂×C₄	—	178	2	no	—
274	C₂×C₄	—	178	2	no	—
275	C₂×C₄	#15667924	1	2	yes	—
276	C₂×C₄	#15667924	1	2	yes	—
277	C₂×C₄	#15667930	1	2	yes	—
278	C₂×C₄	#15667940	1	2	yes	—
279	C₂×C₄	C₂×C₈×D₈₉	1	1	yes	characteristic
280	C₂×C₄	C₂×#13083557	1	1	yes	characteristic
281	C₂×C₄	#15668034	1	2	yes	—
282	C₂×C₄	#15668038	1	1	yes	characteristic
283	C₂×C₄	#15668043	1	1	yes	characteristic
284	C₈	—	4	4	no	—
285	C₈	—	712	2	no	—
286	C₂²	C₂×#15667924	1	1	yes	characteristic
287	C₂²	#19151571	1	1	yes	characteristic
288	C₂²	C₂×#15667930	1	1	yes	characteristic
289	C₂²	#19151595	1	1	yes	characteristic
290	C₂²	#19151597	1	1	yes	characteristic
291	C₂²	C₂×#15667940	1	1	yes	characteristic
292	C₂²	#19151614	1	1	yes	characteristic
293	C₄	—	2	2	no	—
294	C₄	—	2	1	no	—
295	C₄	—	2	1	no	—
296	C₄	—	2	2	no	—
297	C₄	—	2	2	no	—
298	C₄	—	2	2	no	—
299	C₄	—	2	2	no	—
300	C₄	—	2	1	no	—
301	C₄	—	2	1	no	—
302	C₄	—	356	2	no	—
303	C₄	—	356	2	no	—
304	C₄	—	712	1	no	—
305	C₄	—	712	1	no	—
306	C₂	#26018575	1	1	yes	characteristic
307	C₂	#26018635	1	1	yes	characteristic
308	C₂	#26018637	1	1	yes	characteristic
309	C₂	#26018775	1	1	yes	characteristic
310	C₂	#26018830	1	1	yes	characteristic
311	C₂	#26018853	1	1	yes	characteristic
312	C₂	#26018950	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#13083551
commutator	C₂×C₄×C₈₉	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₄×C₄⋊C₈×C₈₉	C₂
socle, prime radical	C₂³×C₈₉	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	C₄×C₄⋊C₈	D₈₉
hypocenter	C₈₉	#7585
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄×C₄⋊C₈

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
25. #15668004	C₈	semidirect
162. C₈₉	#7585	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₄×#15668004	2
#26020395	2
#26018076	2
#26018074	2
C₄×C₄⋊C₈×C₈₉	2
#7585	89